Problems in Fluid Dynamics and Elasticity

拟研究的课题如下：1。液体射流的破裂：本研究将从数学上分析牛顿流体和粘弹性液体射流的毛细破裂。聚合物添加剂在抑制破碎方面的作用是众所周知的，并在自然界（蜘蛛网）和技术上（控制液体喷洒时的液滴大小）得到了开发。本研究将集中于分手的渐近性，并比较各种本构理论。粘弹性流动的稳定性：在过去的十年里，弹性不稳定性得到了广泛的研究。这种不稳定性常常对聚合物的加工条件造成限制。在复杂流动不稳定性的数值研究中，连续谱的分辨率差是一个重大的挑战。研究的目的是利用分析技术来确定这些连续光谱的位置。弹性介质的形状控制：弹性介质由能提供给定类型应力的致动器控制。这项研究将调查这样的控制能否达到给定的介质形状的问题。两种流体流动的数值模拟：本研究将继续进行两种流体流动的数值模拟工作。主要研究以下三个问题：a)由于毛细波的作用，在撞击液滴上形成“阶梯”结构。b)当表面活性剂存在时，液滴在剪切作用下破碎时的倾斜流动。c)正应力对粘弹性液滴剪切破碎的影响。提出的研究解决了在聚合物液体的加工和使用中出现的基本问题，弹性固体变形的控制以及通过剪切混合形成乳液。提出的主题包括在射流破碎中聚合物流体的各种模型的定性行为的比较，聚合物流动的稳定性，在给定类型的控制下弹性介质的形状可以实现的问题，以及两种流体流动的数值方法的发展。

Research is proposed on the following topics:1. Breakup of liquid jets: This research will mathematically analyze the capillary breakup of jets of Newtonian and viscoelastic liquids. The effect of polymer additives in suppressing breakup is well known and exploited both in nature (spider webs) and in technology (control of drops sizes in spraying of liquids). The research will concentrate on the asymptotics of breakup and compare various constitutive theories.2. Stability of viscoelastic flows: Elastic instabilities have been investigated extensively over the past ten years. Such instabilities often pose limits for polymer processing conditions. In the numerical investigation of instabilities in complex flows, continuous spectra which are poorly resolved pose a significant challenge. The research is aimed at using analytical techniques to determine the location of these continuous spectra.3. Shape control in elasticity: An elastic medium is controlled by actuators which can supply a stress of a given type. The research will investigate the question whether such controls can achieve a given shape of the medium.4. Numerical simulation of two-fluid flows: The research will continue work on simulation of flows of two liquids using a volume of fluid code. The following three problems will be investigated: a) Formation of ``staircase" structures on impacting drops due to capillary waves. b) Tipstreaming in drop breakup under shear when surfactants are present. c) Effect of normal stresses on breakup of viscoelastic drops under shear.The proposed research addresses fundamental issues which arise in the processing anduse of polymeric liquids, the control of deformations of elastic solids and theformation of emulsions by shear mixing. The proposed topics include a comparison ofthe qualitative behavior various models of polymeric fluids in jet breakup, thestability of polymeric flows, the question which shapes of an elastic medium canbe achieved by a given type of control, and the development of numerical methods fortwo-fluid flows.